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GATE 2022 EE – Question 38

Control Systems · Stability analysis using Routh-Hurwitz and Nyquist criteria, Bode plots, Root loci · 2 marks · Multiple choice

The open loop transfer function of a unity gain negative feedback system is given as

$$G(s)=\frac{1}{s(s+1)}.$$

The Nyquist contour in the $s$-plane encloses the entire right half plane and a small neighbourhood around the origin in the left half plane, as shown in the figure below. The number of encirclements of the point $(-1+j0)$ by the Nyquist plot of $G(s)$, corresponding to the Nyquist contour, is denoted as $N$. Then $N$ equals to

Diagram for GATE 2022 EE question 38
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Correct answer: (B) 1

Explanation

The contour is clockwise and, because it bulges into the left half plane around the origin, it encloses the open-loop pole at $s=0$, so $P=1$. The closed-loop characteristic equation $s^2+s+1=0$ is stable, so $Z=0$. The number of clockwise encirclements is $N=Z-P=-1$, i.e. the point $-1+j0$ is encircled once (counter-clockwise), so the magnitude is 1.